Volume Growth and General Rate Quantization on Grassmann Manifolds

The Grassmann manifold Gn,p (L) is the set of all p-dimensional planes (through the origin) in the n-dimensional Euclidean space Lⁿ, where L is either R or C. This paper considers an unequal dimensional quantization in which a source in Gn,q (L) is quantized through a code in Gn,p (L), where p and q are not necessarily the same. The analysis for unequal dimensional quantization is based on the volume of a metric ball in Gn,q (L) whose center is in Gn,p (L). Our chief result is to show that as n, p, q and the square radius approach infinity with constant ratios, the volume of a metric ball "grows" as exp (-n²V (1 + o (1))) for a computable constant V ges 0. This result is stronger than our previous volume formula which is only valid when the radius is at most one. The tools behind the present result include large deviation techniques and equilibrium measure ideas from potential theory. Based on the volume growth formula, the rate distortion tradeoff is precisely quantified in our asymptotic region. Finally, we prove that random codes are asymptotically optimal in probability.